High School

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Calculer chaque expression et donner le résultat sous forme d'une fraction plus simple.

[tex]\[

\begin{array}{l}

A=\left(\frac{-3}{5}\right) \times\left(\frac{-30}{18}\right) \times\left(\frac{12}{27}\right) \\

B=\frac{2}{3} \times\left(\frac{-3}{4}\right) \times\left(\frac{-4}{5}\right) \times \frac{5}{8} \\

C=\left(-\frac{20}{50}\right) \times\left(\frac{-40}{30}\right) \times\left(\frac{-25}{-8}\right) \\

D=\left(-\frac{18}{-12}\right) \times\left(\frac{-75}{45}\right) \times\left(\frac{6}{-15}\right)

\end{array}

\][/tex]

Answer :

Let's solve each of these expressions step-by-step and simplify them to the simplest fraction form.

### Expression A
[tex]\[ A = \left(\frac{-3}{5}\right) \times \left(\frac{-30}{18}\right) \times \left(\frac{12}{27}\right) \][/tex]

1. Multiply the numerators:
[tex]\((-3) \times (-30) \times 12 = 1080\)[/tex]

2. Multiply the denominators:
[tex]\(5 \times 18 \times 27 = 2430\)[/tex]

3. Form the fraction:
[tex]\(\frac{1080}{2430}\)[/tex]

4. Simplify the fraction:
We can divide both the numerator and the denominator by their greatest common divisor, which is 270.

[tex]\[\frac{1080 \div 270}{2430 \div 270} = \frac{4}{9}\][/tex]

### Expression B
[tex]\[ B = \frac{2}{3} \times \left(\frac{-3}{4}\right) \times \left(\frac{-4}{5}\right) \times \frac{5}{8} \][/tex]

1. Multiply the numerators:
[tex]\(2 \times (-3) \times (-4) \times 5 = 120\)[/tex]

2. Multiply the denominators:
[tex]\(3 \times 4 \times 5 \times 8 = 480\)[/tex]

3. Form the fraction:
[tex]\(\frac{120}{480}\)[/tex]

4. Simplify the fraction:
Dividing both the numerator and the denominator by their greatest common divisor, which is 120.

[tex]\[\frac{120 \div 120}{480 \div 120} = \frac{1}{4}\][/tex]

### Expression C
[tex]\[ C = \left(-\frac{20}{50}\right) \times \left(\frac{-40}{30}\right) \times \left(\frac{-25}{-8}\right) \][/tex]

1. Multiply the numerators:
[tex]\((-20) \times (-40) \times (-25) = -20000\)[/tex]

2. Multiply the denominators:
[tex]\(50 \times 30 \times (-8) = -12000\)[/tex]

3. Form the fraction:
[tex]\(\frac{-20000}{-12000}\)[/tex]

4. Simplify the fraction:
The negative signs cancel out, so we can divide the numbers by 4000.

[tex]\[\frac{20000 \div 4000}{12000 \div 4000} = \frac{5}{3}\][/tex]

### Expression D
[tex]\[ D = \left(-\frac{18}{-12}\right) \times \left(\frac{-75}{45}\right) \times \left(\frac{6}{-15}\right) \][/tex]

1. Simplifying each fraction before multiplication:
[tex]\(\frac{-18}{-12} = \frac{18}{12} = \frac{3}{2}\)[/tex]

[tex]\(\frac{-75}{45} = \frac{-5}{3} = -\frac{5}{3}\)[/tex]

[tex]\(\frac{6}{-15} = -\frac{2}{5}\)[/tex]

2. Multiply the numerators and denominators:
[tex]\(3 \times (-5) \times (-2) = 30\)[/tex]

[tex]\(2 \times 3 \times 5 = 30\)[/tex]

3. Form the fraction:
[tex]\(\frac{30}{30} = 1\)[/tex]

So, the simplified results for the expressions are:

- [tex]\(A = \frac{4}{9}\)[/tex]
- [tex]\(B = \frac{1}{4}\)[/tex]
- [tex]\(C = \frac{5}{3}\)[/tex]
- [tex]\(D = 1\)[/tex]

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